I'm studying Lagrangian mechanics, but I'm a little bit upset because when dealing with Lagrange's equations, we mostly consider conservative systems. If the system is non conservative they are very brief by saying that 'sometimes' there exist a velocity dependent potential $U(q,\dot{q},t)$ such that the generalized force $Q_j$ of the standard system can be written in terms of this potential. $$Q_j =\frac{d}{dt}\left(\frac{\partial U}{\partial \dot{q}_j}\right)-\frac{\partial U}{\partial q_j}$$ They give as example, charged particles in a static EM field.

  1. But my question is, if we can find this velocity dependent potential for any generalized force?

  2. If not, we can't use Lagrangian mechanics?

  1. No, (generalized) velocity dependent potentials $U(q,\dot{q},t)$ do not exist for all (generalized) forces $Q_j$. See e.g. this Phys.SE post.

  2. Even if no variational formulation exists, one may still consider Lagrange equations, cf. e.g. this Phys.SE post.

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